Democracy of quasi-greedy bases in $p$-Banach spaces with applications to the efficiency of the TGA in the Hardy spaces $H_p(\mathbb{D}^d)$

TL;DR

This paper investigates the conditions under which quasi-greedy bases in non-locally convex $p$-Banach spaces are democratic, revealing that all such bases in $H_p( ext{D})$ for $p<1$ are democratic, but not in higher dimensions.

Contribution

The paper introduces new methods specific to non-locally convex quasi-Banach spaces to determine when quasi-greedy bases are democratic, solving an open problem for Hardy spaces.

Findings

All quasi-greedy bases of $H_p( ext{D})$ for $0<p<1$ are democratic.

No quasi-greedy basis of $H_p( ext{D}^d)$ for $d extgreater 1$ is democratic.

Applications to other functional spaces are also discussed.

Abstract

We use new methods, specific of non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a $p$-Banach space for $0<p<1$ are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy space $H_p(\mathbb{D})$ for $0<p<1$ are democratic while, in contrast, no quasi-greedy basis of $H_p(\mathbb{D}^d)$ for $d\ge 2$ is, solving thus a problem that was raised in [F. Albiac, J. L. Ansorena, and P. Wojtaszczyk, \textit{Quasi-greedy bases in $\ell_p$ ($0<p<1$) are democratic}, J. Funct. Anal. \textbf{280} (2021), no. 7, 108871, 21]. Applications of our results to other spaces of interest both in functional analysis and approximation theory are also provided.